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Your data matches 26 different statistics following compositions of up to 3 maps.
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Mp00252: Permutations restrictionPermutations
St000054: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 3
[3,1,4,2] => [3,1,2] => 3
[3,2,1,4] => [3,2,1] => 3
[3,2,4,1] => [3,2,1] => 3
[3,4,1,2] => [3,1,2] => 3
[3,4,2,1] => [3,2,1] => 3
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 3
[4,3,2,1] => [3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Matching statistic: St000382
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> [1] => 1
[2,1] => [1] => [1,0]
=> [1] => 1
[1,2,3] => [1,2] => [1,0,1,0]
=> [1,1] => 1
[1,3,2] => [1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1,3] => [2,1] => [1,1,0,0]
=> [2] => 2
[2,3,1] => [2,1] => [1,1,0,0]
=> [2] => 2
[3,1,2] => [1,2] => [1,0,1,0]
=> [1,1] => 1
[3,2,1] => [2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[4,2,5,1,3,6,7,8,9,10] => [4,2,5,1,3,6,7,8,9] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? => ? = 4
[5,2,6,7,1,3,4,8,9,10] => [5,2,6,7,1,3,4,8,9] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? => ? = 5
[6,3,2,7,8,1,4,5,9,10] => [6,3,2,7,8,1,4,5,9] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? = 6
[7,4,3,2,8,9,1,5,6,10] => [7,4,3,2,8,9,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? => ? = 7
Description
The first part of an integer composition.
Matching statistic: St000011
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> [1,0]
=> 1
[2,1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 7
[5,6,7,8,10,1,2,3,4,9] => [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[10,6,7,8,9,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,4,2,5,3,7,6,9,8] => [1,4,2,5,3,7,6,8] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[3,4,1,2,5,6,7,8,9] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
[4,5,6,1,2,3,7,8,9] => [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[4,2,5,1,3,6,7,8,9] => [4,2,5,1,3,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[5,2,6,7,1,3,4,8,9] => [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5
[5,3,2,6,1,4,7,8,9] => [5,3,2,6,1,4,7,8] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[6,3,2,7,8,1,4,5,9] => [6,3,2,7,8,1,4,5] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6
[6,4,3,2,7,1,5,8,9] => [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[3,4,1,2,5,6,7,8,9,10] => [3,4,1,2,5,6,7,8,9] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
[4,5,6,1,2,3,7,8,9,10] => [4,5,6,1,2,3,7,8,9] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[4,2,5,1,3,6,7,8,9,10] => [4,2,5,1,3,6,7,8,9] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[5,6,7,8,1,2,3,4,9,10] => [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[5,2,6,7,1,3,4,8,9,10] => [5,2,6,7,1,3,4,8,9] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5
[5,6,3,4,1,2,7,8,9,10] => [5,6,3,4,1,2,7,8,9] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[5,3,2,6,1,4,7,8,9,10] => [5,3,2,6,1,4,7,8,9] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[6,2,7,8,9,1,3,4,5,10] => [6,2,7,8,9,1,3,4,5] => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[6,7,3,4,8,1,2,5,9,10] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6
[6,3,2,7,8,1,4,5,9,10] => [6,3,2,7,8,1,4,5,9] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6
[6,4,7,2,5,1,3,8,9,10] => [6,4,7,2,5,1,3,8,9] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[6,4,3,2,7,1,5,8,9,10] => [6,4,3,2,7,1,5,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[7,4,3,2,8,9,1,5,6,10] => [7,4,3,2,8,9,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[7,5,3,8,2,6,1,4,9,10] => [7,5,3,8,2,6,1,4,9] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7
[7,5,4,3,2,8,1,6,9,10] => [7,5,4,3,2,8,1,6,9] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[8,6,9,4,7,2,5,1,3,10] => [8,6,9,4,7,2,5,1,3] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 8
[8,6,4,3,9,2,7,1,5,10] => [8,6,4,3,9,2,7,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 8
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> 2 = 1 + 1
[2,1] => [1] => [1,0]
=> 2 = 1 + 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[2,3,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[4,2,3,5,6,7,8,9,1] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[5,6,7,8,10,1,2,3,4,9] => [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 1
[7,8,9,10,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[10,7,8,9,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[10,6,7,8,9,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[10,7,8,9,4,5,6,1,2,3] => [7,8,9,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[1,10,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,11,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[3,4,1,2,5,6,7,8,9] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,5,6,1,2,3,7,8,9] => [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[4,2,5,1,3,6,7,8,9] => [4,2,5,1,3,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[5,2,6,7,1,3,4,8,9] => [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 5 + 1
[5,6,3,4,1,2,7,8,9] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
[5,3,2,6,1,4,7,8,9] => [5,3,2,6,1,4,7,8] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[6,4,3,2,7,1,5,8,9] => [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[3,4,1,2,5,6,7,8,9,10] => [3,4,1,2,5,6,7,8,9] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,5,6,1,2,3,7,8,9,10] => [4,5,6,1,2,3,7,8,9] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,2,5,1,3,6,7,8,9,10] => [4,2,5,1,3,6,7,8,9] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[5,6,7,8,1,2,3,4,9,10] => [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 1
[5,2,6,7,1,3,4,8,9,10] => [5,2,6,7,1,3,4,8,9] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
[5,6,3,4,1,2,7,8,9,10] => [5,6,3,4,1,2,7,8,9] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[5,3,2,6,1,4,7,8,9,10] => [5,3,2,6,1,4,7,8,9] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[6,2,7,8,9,1,3,4,5,10] => [6,2,7,8,9,1,3,4,5] => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 + 1
[6,7,3,4,8,1,2,5,9,10] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[6,3,2,7,8,1,4,5,9,10] => [6,3,2,7,8,1,4,5,9] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[6,4,7,2,5,1,3,8,9,10] => [6,4,7,2,5,1,3,8,9] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[6,4,3,2,7,1,5,8,9,10] => [6,4,3,2,7,1,5,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 6 + 1
[6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 6 + 1
[7,8,9,4,5,6,1,2,3,10] => [7,8,9,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[7,4,3,2,8,9,1,5,6,10] => [7,4,3,2,8,9,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 7 + 1
[7,5,3,8,2,6,1,4,9,10] => [7,5,3,8,2,6,1,4,9] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 + 1
[7,5,4,3,2,8,1,6,9,10] => [7,5,4,3,2,8,1,6,9] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 7 + 1
[7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 + 1
[8,6,4,3,9,2,7,1,5,10] => [8,6,4,3,9,2,7,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 8 + 1
Description
The position of the first down step of a Dyck path.
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000971: Set partitions ⟶ ℤResult quality: 90% values known / values provided: 98%distinct values known / distinct values provided: 90%
Values
[1,2] => [1] => [1,0]
=> {{1}}
=> 1
[2,1] => [1] => [1,0]
=> {{1}}
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,9},{6},{7},{8}}
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 2
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 2
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 2
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 2
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 2
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 3
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 2
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> ? = 2
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> ? = 2
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 2
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 8
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,8},{4,5,7},{6},{9}}
=> ? = 6
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,9},{4,5,8},{6,7}}
=> ? = 7
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 2
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 2
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 1
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 3
[9,10,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,10,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,10,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,10,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,5,10,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,5,4,10,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,5,4,3,10,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,5,4,3,2,10,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
Description
The smallest closer of a set partition. A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers. In other words, this is the smallest among the maximal elements of the blocks.
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 97%distinct values known / distinct values provided: 70%
Values
[1,2] => [1] => [1,0]
=> 1
[2,1] => [1] => [1,0]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> ? = 7
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of D.
Matching statistic: St000026
Mp00252: Permutations restrictionPermutations
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000026: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 97%distinct values known / distinct values provided: 70%
Values
[1,2] => [1] => [.,.]
=> [1,0]
=> 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,4,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,1,2,4] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[3,1,4,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[4,2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[4,3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 10
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],[.,.]]
=> ?
=> ? = 8
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.],.]
=> ?
=> ? = 9
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,.]]]
=> ?
=> ? = 7
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The position of the first return of a Dyck path.
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00137: Dyck paths to symmetric ASMAlternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 90% values known / values provided: 93%distinct values known / distinct values provided: 90%
Values
[1,2] => [1] => [1,0]
=> [[1]]
=> 1
[2,1] => [1] => [1,0]
=> [[1]]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 5
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 4
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 4
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 3
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 3
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 5
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 5
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 7
Description
The column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by \binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!}, see [2].
Matching statistic: St000297
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
St000297: Binary words ⟶ ℤResult quality: 90% values known / values provided: 93%distinct values known / distinct values provided: 90%
Values
[1,2] => [1] => [1,0]
=> 10 => 1
[2,1] => [1] => [1,0]
=> 10 => 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 1010 => 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 1010 => 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 1100 => 2
[2,3,1] => [2,1] => [1,1,0,0]
=> 1100 => 2
[3,1,2] => [1,2] => [1,0,1,0]
=> 1010 => 1
[3,2,1] => [2,1] => [1,1,0,0]
=> 1100 => 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 111000 => 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 111000 => 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 111000 => 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 111000 => 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 10111000 => 1
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> 11101010010100 => ? = 3
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 11111000010100 => ? = 5
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 11110001010100 => ? = 4
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 11101001010100 => ? = 3
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 11110001010100 => ? = 4
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => ? = 3
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 2
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 11111100100000 => ? = 6
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 11101010101000 => ? = 3
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 11101010101000 => ? = 3
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 6
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 6
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 11111010100000 => ? = 5
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 11111010100000 => ? = 5
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 6
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 6
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 11111101000000 => ? = 6
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? = 7
Description
The number of leading ones in a binary word.
Matching statistic: St000738
Mp00252: Permutations restrictionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 70% values known / values provided: 93%distinct values known / distinct values provided: 70%
Values
[1,2] => [1] => [1,0]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1] => [1] => [1,0]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[1,3,2] => [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[2,1,3] => [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1] => [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,1,2] => [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[3,2,1] => [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,3,5,7,10,12],[4,6,8,9,11,13,14]]
=> ? = 3 + 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[1,2,3,4,5,10,12],[6,7,8,9,11,13,14]]
=> ? = 5 + 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,4,8,10,12],[5,6,7,9,11,13,14]]
=> ? = 4 + 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,5,8,10,12],[4,6,7,9,11,13,14]]
=> ? = 3 + 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,4,8,10,12],[5,6,7,9,11,13,14]]
=> ? = 4 + 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12],[4,5,7,9,11,13,14]]
=> ? = 3 + 1
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? = 2 + 1
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,6,9],[7,8,10,11,12,13,14]]
=> ? = 6 + 1
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9,11],[4,6,8,10,12,13,14]]
=> ? = 3 + 1
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9,11],[4,6,8,10,12,13,14]]
=> ? = 3 + 1
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 6 + 1
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 6 + 1
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7,9],[6,8,10,11,12,13,14]]
=> ? = 5 + 1
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7,9],[6,8,10,11,12,13,14]]
=> ? = 5 + 1
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 6 + 1
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 6 + 1
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 6 + 1
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St000740The last entry of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000051The size of the left subtree of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000260The radius of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001816Eigenvalues of the top-to-random operator acting on a simple module.